Hmm, this hasn't really fully clicked with me yet (so take what I am saying with a grain of salt, I'd be glad if anyone could clarify/prove me wrong), but speed is relative.
If your station moves fast from the perspective of earth, earth moves fast from the perspective of the station. So, from the perspective of earth, time moves slower on the station, and from the perspective of the station, time moves slower on earth.
If you were to travel back and forth, that would even out (something about having to accelerate in one direction, then another). There is no absolute time reference in the universe.
That can work, however, if you spend some time inside a strong gravity field.
The time delta math (Sqrt(1-v^2/c^2)) puts us at 3 earth days per two space days.
The vacationing was more cheeky, in that in this case a 2 week vacation would appear 3 weeks on earth. If there would be some way to quickly switch between modes than 1 week, you would spend more self-referential time doing something than your home clock would count.
A couple of high school sweethearts promise they'll love each other forever, but there's a war on and the boy quickly gets recruited to fight at the edge of the solar system (I remember a scene where they stand at some railroad tracks and watch a monumental silver space craft fly overhead, one of the battle cruisers), but before he gets there the boy and girl continue their relationship as pen pals, at first sending messages back and forth from mars base (45 minute delay, but not aging any faster), but as the boy travels closer to the front of the war, his velocity increases and he ages less than the girl waiting for him back on earth, and of course more time passes by between messages.
Eventually it's the boys 20something birthday and he still has love for the girl, while the girl has already grown up and moved on with her life.
Edit: hah, figures my bias would mix the genders up, the girl went off to fight the war.
If you mean the actual Earth and a free-fall orbit, the orbital speed of something in low earth orbit, like the ISS, about 8 km/sec, is as fast as anything can orbit the earth.
To get really fast orbital speeds (as in, speeds that are any significant fraction of the speed of light), you need objects much, much more compact than the Earth, objects whose actual size is not that much larger than the size of a black hole with the same mass. The only non-black hole objects we know of that are anywhere close to that compact are neutron stars.
My standard model of “Thing going really fast” has been some nuclear powered craft that just shoots out into the abyss, but after a bit of googling apparently our fastest craft is/will be Juno, which is staying within the solar system. This will apparently go 250,000km/hr or 0.000232c partially based on Jupiter’s gravity.
It is interesting to think about just “living fast”, where our lives are still only ~80 years, but the time elapsed on earth is on the order of many multiples of that. We would exist in a way that we orbit the earth at a much larger fraction of light speed, while still allowing us to interact with the earth’s resources on normal time scales. Imagine taking a vacation back to earth for a year, but popping back to the space station where it has only been a few hours.
Maybe I’ll write a novel on that or find one already written, there seems to be a lot of depth in exploring that.
[0] https://www.missionjuno.swri.edu/earth-flyby/story/how-fast-...